Book for brushing up Maths

[Tsotne]

i just started viewing gilbert strang's video lectures on MIT OCW and am finding them quite useful .... will the 35 lectures on linear algebra take care of the course ?

Yes it should be quite enough. Including homeworks you'll be doing, during the time you gain quite a lot knowledge you need to move on to the next steps you mentioned above.

 

[achilles]

ya , i will check on prof shreve's books and sheldon ross' probability ones ...... maybe after viewing them online i could order a hard copy.

 

[Tsotne]

I also got interested in Sheldon Ross's book. I'll search for it too. Good Luck

 

[myampol]

I wouldn't recommend starting with Ross's "Stochastic Processes" unless you are already comfortable with the material in his other books.

Depending upon your background, you would be much better off starting with either Ross's "First Course in Probability" (weak background) or "Introduction to Probability Models" (medium background).

 

[cgorac]

i just started viewing gilbert strang's video lectures on MIT OCW and am finding them quite useful .... will the 35 lectures on linear algebra take care of the course ?

Video lectures cover "Introduction to Linear Algebra" book almost completely, and mostly in order (only applications chapter material is scattered throughout the course) - you could check for book sections matching video lectures in Readings section of OCW 18.06 site. The book itself is rather comprehensive, so once you complete through the book and video lectures, you should be pretty good regarding linear algebra (but don't forget you really need to work on book problem, homework and exams in order to master the material properly).

 

[bigbadwolf]

Video lectures cover "Introduction to Linear Algebra" book almost completely, and mostly in order (only applications chapter material is scattered throughout the course) - you could check for book sections matching video lectures in Readings section of OCW 18.06 site. The book itself is rather comprehensive, so once you complete through the book and video lectures, you should be pretty good regarding linear algebra (but don't forget you really need to work on book problem, homework and exams in order to master the material properly).

I'm looking at my copy of Strang's "Linear Algebra and its Applications," which you wrote about in an earlier post (I have the 3rd edition). Cayley-Hamilton has been relegated to the exercises; there's no discussion I can see of the minimum polynomial; the discussion of the Spectral Theorem isn't quite clear. There are probably other things I will catch as well if I look closely. It's possible some of these things have been cleared up in the 4th edition. But I feel there's a case to be made for a more orthodox treatment of the subject as can be found, for example, in the inexpensive Schaum series book (by Lipschutz?).

 

[Tsotne]

Those blue books Baruch advised for stochastic mathematics seem to be good but regrettably impossible to read on parallel with other similar books.

 

[rajanS]

you could try steven roman. his book covers the probability background and even relevant chapters on sigma algebras etc. its a very elementary book and assumes background in only linear algebra. he covers 3 chapters in discrete to continuous and its quite amazing how he doesn't even use calculus to cover the continuous concepts.

 

[cgorac]

I'm looking at my copy of Strang's "Linear Algebra and its Applications," which you wrote about in an earlier post (I have the 3rd edition). Cayley-Hamilton has been relegated to the exercises; there's no discussion I can see of the minimum polynomial; the discussion of the Spectral Theorem isn't quite clear. There are probably other things I will catch as well if I look closely. It's possible some of these things have been cleared up in the 4th edition. But I feel there's a case to be made for a more orthodox treatment of the subject as can be found, for example, in the inexpensive Schaum series book (by Lipschutz?).

Strang books are certainly not perfect. On the other side if you know about the subject, then most of the time you'll find something you think it's important missing from any kind of book - for example, Lipschutz book you mentioned above is not discussing (as far as I can see from book index on Amazon) SVD at all. But we're talking introductory books on linear algebra here; Strang books seem to be doing job well for students of MIT, and number of other unis, and are accompanied with video lectures and courses materials, which makes them worth consideration in my opinion.

 

[Tsotne]

you could try steven roman. his book covers the probability background and even relevant chapters on sigma algebras etc. its a very elementary book and assumes background in only linear algebra. he covers 3 chapters in discrete to continuous and its quite amazing how he doesn't even use calculus to cover the continuous concepts.

Yes sigma algebra background would be helpful before starting stochastic math...

 

[bigbadwolf]

Strang books are certainly not perfect.

I like Strang's books -- particularly "Computational Science and Engineering."

 

[Devdeep]

While these books will provide you some background on elementary probability and on various stochastic processes, none of them will give you what you need to know to tackle the "really hard stuff" that makes up Stochastic Calculus. For this you also need to cover that part of mathematics called "Real Analysis", i.e., "Theory of Functions of a Real Variable." There are many books which address this topic, which most people find far more challenging than simple non-measure-theoretic probability and stochastic processes which are addressed by Ross's books.

One set of books which you might find quite helpful in getting the necessary background that you would need in order to understand the advanced stuff that you are interested in is a pair of books by Grimmett and Stirzaker. Their textbook is called "Probability and Random Processes" (be sure to get the third edition, from 2001) and their companion exercise book is called "One Thousand Exercises in Probability."
http://www.amazon.com/Probability-Random-Processes-Geoffrey-Grimmett/dp/0198572220
http://www.amazon.com/Thousand-Exercises-Probability-Geoffrey-Grimmett/dp/0198572212

Another book which you might find helpful is "An Introduction to Measure-Theoretic Probability" by George Roussas.
http://www.amazon.com/Introduction-Measure-theoretic-Probability-George-Roussas/dp/0125990227

Thanks for the informative posts.

Do you have any comments about

1) Elements of the Theory of Functions and Functional Analysis by A. N. Kolmogorov, S. V. Fomin

and

2) Principles of Mathematical Analysis by Walter Rudin

for understanding Real Analysis?

 

[snipez]

Baby Rudin (book 2) is the standard undergraduate real analysis text. The exposition is terse and elegant, so if you haven't done any analysis before, be prepared to work hard or find a different text. Other texts you may wish to consider are Tom Apostol's Mathematical Analysis or Max Rosenlicht's Introduction to Analysis.

You would really need the contents of the first 7 chapters of Baby Rudin to tackle Kolmogorov and Fomin (K&F). K&F covers measure theory by first constructing the Lebesgue measure on plane sets first before generalizing the idea to other sets. This approach is perhaps more intuitive than more modern treatments, but Big Rudin (Real and Complex Analysis) and Royden's Real Analysis are the standard treatments.

 

[Devdeep]

Baby Rudin (book 2) is the standard undergraduate real analysis text. The exposition is terse and elegant, so if you haven't done any analysis before, be prepared to work hard or find a different text. Other texts you may wish to consider are Tom Apostol's Mathematical Analysis or Max Rosenlicht's Introduction to Analysis.

You would really need the contents of the first 7 chapters of Baby Rudin to tackle Kolmogorov and Fomin (K&F). K&F covers measure theory by first constructing the Lebesgue measure on plane sets first before generalizing the idea to other sets. This approach is perhaps more intuitive than more modern treatments, but Big Rudin (Real and Complex Analysis) and Royden's Real Analysis are the standard treatments.

Thanks a bunch~!

** Max Rosenlicht's Introduction to Analysis - LIKE LIKE...

 

[bigbadwolf]

Baby Rudin (book 2) is the standard undergraduate real analysis text. The exposition is terse and elegant, so if you haven't done any analysis before, be prepared to work hard or find a different text. Other texts you may wish to consider are Tom Apostol's Mathematical Analysis or Max Rosenlicht's Introduction to Analysis.

These are all okay texts, with Rosenlicht having the advantage of being published by Dover (I think) and hence inexpensive. Baby Rudin is too expensive for what it is -- unless one can get hold of one of the Asian reprints. Not sure Apostol is in print (haven't bothered checking). There are at least a dozen texts I can think of that are better than the dated Rudin. My favorite among them is Sohrab's Basic Real Analysis, which covers not only undergrad analysis and general topology but also introduces Banach and Hilbert spaces, the Lebesgue integral, with a final chapter devoted to general measure and measure-theoretic probability. It is well organised and the theorems are both motivated by examples and elegantly proved. This is a book for the connoisseur. There are simpler books published by Springer UTM and GTM, and by the MAA.

 

[Devdeep]

There are at least a dozen texts I can think of that are better than the dated Rudin. My favorite among them is Sohrab's Basic Real Analysis, ... There are simpler books published by Springer UTM and GTM, and by the MAA.

Thanks!

Can you name your favs from the UTM, GTM series?

 

[menovive]

I just started off with Calculus I, II, and III by Jerrold Marsden; First Course by Ross; Linear Algebra by Axler !! -> All undergrad Texts: Available for Cheap Rates at DealOz /Half.com

 

[bigbadwolf]

Can you name your favs from the UTM, GTM series?

If I were teaching analysis to a class of undergrad sophomores without much mathematical maturity I would use

1) Real Analysis by Howie, published by Springer

and/or

2) A Radical Approach to Real Analysis by Bressoud, published by the Mathematical Association of America.

If the class was a trifle more mature, either

3) Undergraduate Analysis, by Abbott, pub. by Spinger,

or

4) Real Mathemaical Analysis, by Pugh, pub. Springer.

Postscript: The previous poster mentioned Marsden, which reminds me of Marsden's "Elementary Classical Analysis." This is another phenomenal book, roughly at the level of Apostol, but to my mind written in a more limpid style. I don't kniw if it's in print anymore.

 

[Devdeep]

If I were teaching analysis to a class of undergrad sophomores without much mathematical maturity I would use

1) Real Analysis by Howie, published by Springer

and/or

2) A Radical Approach to Real Analysis by Bressoud, published by the Mathematical Association of America.

If the class was a trifle more mature, either

3) Undergraduate Analysis, by Abbott, pub. by Spinger,

or

4) Real Mathemaical Analysis, by Pugh, pub. Springer.

Postscript: The previous poster mentioned Marsden, which reminds me of Marsden's "Elementary Classical Analysis." This is another phenomenal book, roughly at the level of Apostol, but to my mind written in a more limpid style. I don't kniw if it's in print anymore.

Thanks!

 

[bigbadwolf]

I just found one more introductory book on my shelf:

5) Introductory Real Analysis, by Dangello and Seyfried, published by Houghton-Mifflin.

Haven't checked to see if it's still in print. Along with the books by Howie and Bressoud above, it's an excellent introduction for students who have been through the calculus sequence but don't quite understand the need for rigorous proofs or understand the need for coherence in subject matter.